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Math 115 Lab 7 Solutions

1. Consider transformation T : R3 R3 : [a b c] 7 [0 2a b].a)Write the matrix A associated to this transformation.

b) Find the image space of A and determine its dimension by finding a basis for it.

c) Find the null space of A and determine its dimension by finding a basis for it.

Answer: a) The matrix associated to T is

A =

0 0 02 0 00 1 0

b) The image space im(A) is

im(A) = {[0 ]T R3}It is a subspace of dimension 2 in R3 and a basis is

~e1 =[0 1 0

]T~e2 =

[0 0 1

]Tc) The null space (A) is

null(A) = {[0 0 ]T R3}In fact 0 0 02 0 0

0 1 0

xyz

= 02xy

=000

{x = 0y = 0

It is a subspace of dimension 1 in R3 and a basis is

~e0 =[0 0 1

]T

2. Consider the following vectors in R4

~u1 =[0 1 2 1]T ~u2 = [1 1 0 2]T ~u3 = [0 0 1 2]T ~u4 = [0 0 1 1]T

a) Show that they are independent.

b) Define ~u5 =[1 0 5 2]T ; is the set {~u1, ~u2, ~u3, ~u4, ~u5} an independent set of vectors? Justify

your answer.

Answer: a) The matrix associated to the system is

A =

0 1 2 11 1 0 20 0 1 20 0 1 1

{R1 R2R4 R4 +R3

1 1 0 20 1 2 10 0 1 20 0 0 3

From its reduced form we see that rank(A) = 4 (det(A) = 3). The rows of A span R4.b) No. In R4 there is set of five vectors which are independent.

3. a) Find a basis for the subspace W spanned by the following vectors in R3

~u1 = [3 1 2]T ~u2 = [1 0 5]T ~u3 = [3 1 2]T ~u4 = [2 1 3]T

b) Find a vector ~v R3 which does not belong W .c) Tell if the set V = {~v R3 : ~v 6W} is a subspace of R3. Justify your answer.

Answer: Just write the corresponding matrix and reduce it.

A =

3 1 21 0 53 1 22 1 3

R2 R2 R1 R4R3 R3 +R2 R4R4 R4 + 23R1

3 1 20 0 00 0 00 1/3 13/3

The vectors ~e1 =

[3 1 2]T and ~e2 = [0 1 13]T form a basis of W .

b) ~v =[0 0 1

]T does not belong toW since it is not a linear combination of ~e1 and ~e2. The dimensionof the subspace span{~e1, ~e2, ~v} is in fact 3.c) No. ~0 6 V .

4. Consider the matrix

A =

3 1 21 2 03 1 2

a) Find the subspace X of R3 spanned by the rows.b) Is A invertible?

Answer: a) By reduction

A =

3 1 21 2 03 1 2

1 2 00 7 20 2 4

The rank of A is clearly 3. The only subspace of R3 which is 3 dimensional is R3 itself.b) The rows are linearly independent, thus A is invertible.

5. Consider the vectors~v1 = [1 1 0]T ~v2 = [0 1 1]T

a) Prove that ~v1 and ~v2 are linearly independent.

b) Find a vector ~w such that {~v1, ~v2, ~w} is a basis of R3.

Answer: a)

~v1 + ~v2 = ~0 [ + ] = [0 0 0] { = 0 = 0

b) Any vector which is not a linear combination of ~v1 and ~v2, for example ~w = [0 2 1]T .

6. Compute the dimension of the space generated by the rows of the following matrix

A =[1 21 0

]

Answer: The rank of the matrix is 2 (the matrix is invertible - rows are independent), so its rowsgenerate all R2.

7. Consider the two vectors in R2

~v1 = [cos sin]T ~v2 = [1 1]T

Discuss the dimension of W = span(~v1, ~v2) for all values of [0, 2pi].

Answer: The associated matrix is

A =[cos sin1 1

]and we have to determine how many independent columns we have.

det(A) = cos+ sin

When det(A) = 0, columns are linearly dependent. This happens when cos = sin or tan = 1.This is true for all {pi4 , 3pi4 }, and then the dimension of W is 1. For 6 {pi4 , 3pi4 } the dimensionis 2.